This study aims to develop constitutive equations for linear thermoelastic analysis of a composite material
having piezoelectric feature and reinforced by arbitrary a fiber family. Fiber-reinforced composite media are
assumed to be of anisotropic nature and are considered to be compressible due to their piezoelectric properties.
Besides, it is assumed that the fiber family is inextensible. In addition, since the composite material is insensitive
to the directional change along the fiber, it is mathematically unaffected by the change B→−B of fiber vector,
so a symmetric a symmetric tensor, which is the outer product of the components of the fiber vector, is defined.
The basis of this work is the equations of electro-thermomechanical equilibrium equations, fiber deformation
geometry and kinematics. The use of constitutive axioms has shown that the stress potential is dependent on the
Green deformation tensor, the fiber distribution tensor, the electric field vector and the absolute temperature, and the heat vector function is dependent on together with these magnitudes the gradient of the temperature field.
Because of the anisotropic nature of the composite media, the stress potential and heat vector functions are found
in approximate theories, and all of the interactions are considered as linear and series expansion is performed.
Field equations are found as a result of substituting the expressions in equilibrium equations and the linear
constitutive equations of symmetric stress, polarization and heat flux vector written in spatial coordinates.
1. Demirtürk S., 2011, “Fiber Takviyeli Piezoelektrik Kompozit Malzemelerin ElektroTermoelastik
Analizi”, Master Tezi, Süleyman Demirel Üniversitesi, Fen Bilimleri
Enstitüsü, Isparta.
2. Öntürk, N., 1993, “İki Fiber Ailesi ile Takviyeli Viskoelastik Kompozit Ortamlarda
Bünye Denklemlerinin Modellenmesi”, Doktora Tezi, Gazi Üniversitesi, Fen Bilimleri
Enstitüsü, Ankara.
3. Erdem, A. Ü., Usal. M. R., Usal. M., 2005, “Keyfi Fiber Takviyeli Viskoelastik
Piezoelektrik Bir Cismin Elektro-Termomekanik Davranışı İçin Matematiksel Bir
Model’’, Gazi Üniv. Müh. Mim. Fak. Dergisi, 20(3) 305-319.
4. Mikata, Y. 2000, “Determination of piezoelectric Eshelby tensor in transversely isotropic
piezoelectric solids’’, Int. J. Eng. Sci., 38, 605-641.
5. Dinzart, F., Sabar, H. 2009, “Electroelastic behavior of piezoelectric composites with
coated reinforcements: Micromechanical approach and applications’’, Int. J. Sol. and St.,
46, 3556–3564.
6. Tong, Z.H., Lo, S.H., Jiang, C.P., Cheung, Y.K. 2008, “An exact solution for the threephase
thermo-electro-magneto-elastic cylinder model and its application to piezoelectricmagnetic
fiber composites’’, Int. J. Sol. and Str., 45, 5205–5219.
7. Timoshenko, S.P., Goodier, J.N., 1970, “Theory of Elasticitiy’’, Mcgraw Hill, 567p.
8. Lubarda, V.A., 2004, “On Thermodynamic Potentials in Linear Thermoelasticity’’, Int. J.
Sol. Str., 41(26), 7377-7398.
9. Usal, M., Hamamcı, B. 2011, “A Mathematical Model for the Linear Constitutive
Equations of a Thermoelastic Composite Continuum Reinforced by Single Family of
Arbitrarily Fiber’’, J. Fac. Eng. Arch. Gazi Univ., 26(2), 315-324.
10. Maugin, A.G., Berezovski, A., 1999, “Material Formulation of Finite-Strain
Thermoelasticity and Applications’’, J. Ther. Stres. 22(4 and 5), 421-449.
11. Kalpakides, K.V., Dascalu, C., 2002, “On the Configurational Force Balance in
Thermamechanics, Proceedings’’, Math., Phy. and Eng. Sci., 458(2028), 3023-3039.
12. Eringen, A.C., 1966, “A Unified Theory of Thermomechanical Materials’’, Int. J. Engng.
Sci., 4, 179-202.
13. Mindlin, R.D., 1972, “Elasticity, Piezoelectricity and Crystal Lattice Dynamics’’, J.
Elasticity, 2, 217-282.
14. Tiersten, H.F., 1971, “On The Nonlinear Equations of Thermoelectro-Elasticity’’, Int. J.
Eng. Sci., 9, 587-604.
15. Nowacki, W., 1975, “Dynamic Problems of Thermoelasticity’’, Noordhoff International
Publishing, Netherlands.
16. Usal, M., Usal, M. R. Kurbanoğlu, C., 2009, “A Mathematical Model for the
Electrothermomechanical Behavior of Viscoelastic Piezoelectric Body Having
Hegzagonal Simetry’’, Sci. and Eng. Comp. Mater., 16, 153-171.
17. Usal M., Usal M.R, Erdem, A.Ü., 2009, “On Magneto-Viscoelastic Behavior of FiberReinforced
Composite Materials Part - I: Anisotropic Matrix Material’’, Sci. and Eng.
Comp. Mater., 16, 41-56.
19. Usal, M., Usal, M.R., and Esendemir, Ü. 2008, “A Continuum Formulation for Fiber -
Reinforced Viscoelastic Composite Materials with Microstructure Part-I: Anisotropic
Matrix Material’’, Sci. and Eng. Comp. Mater., 15(3), 217-234.
20. Usal, M.R, Usal, M., and Esendemir, Ü. 2006, “A Mathematical Model for
Thermomechanical Behavior of Arbitrary Fiber Reinforced Viscoelastic Composites–I’’,
Sci. and Eng. Comp. Mater., 13(4), 291-300.
21. Usal, M. 2010, “A Constitutive Formulation for the Linear Thermoelastic Behavior of
Arbitrary Fiber-Reinforced Composites’’, Mathem. Prob. Eng., (2010), Article ID 404398,
19 pages.
22. Usal, M. 2010, “Formulation of the Linear Constitutive Equations of Thermoelastic
Piezoelectric Materials’’, Electronic Journal of Machine Technologies, 7 (1), 13-30.
23. İnan, M. 1970, “Cisimlerin Mukavemeti’’, II. Baskı, Ofset Matbaacılık Ltd. Şti., İstanbul.
24. Spencer, A.J.M. 1984, “Continuum Theory of the Mechanics of Fibre Reinforced
Composites’’, Spencer, Springer Verlag, 284 p, New York.
25. Usal, M.R. 1994, “A mathematical model for the electro-thermomechanical behaviour of
fiber reinforced elastic dielectric media’’, Ph. D. Thesis, Erciyes University, Institute of
Science and Technology, Kayseri, pp: 108.
26. Hamamcı, B. 2006, “A mathematical model for fiber reinforced thermoelastic materials’’,
Master Thesis, Süleyman Demirel Univ., Institute of Science and Technology, Isparta,
pp: 93.
27. Eringen, A.C., Maugin,G.A., 1990, “Electrodynamics of continua: Foundations and Solid
Media’’, Springer-Verlag, New York.
28. Maugin, G.A., 1991, “Continuum Mechanics of Electromagnetic Solids’’, North –
Holland Series Applied Mathematics and Mechanics, Elsevier Scie. Netherlands.
29. Eringen, A.C., 1980, “Mechanics of Continua’’, Robert E. Krieger Pub. Co., Hungtington,
New York.
30. Şuhubi E.S., 1994, “Continuum mechanics–Introduction’’, İ.T.U., Faculty of Arts and
Sciences Publication. İstanbul.
FİBER TAKVİYELİ PİEZOELEKTRİK KOMPOZİTLERİN ELEKRO-TERMOELASTİK ANALİZİ İÇİN MATEMATİKSEL BİR MODEL
Bu çalışma, fiber takviyeli ve piezoelektrik özelliğe sahip bir kompozit malzemenin lineer elektro-termoelastik
analizi için bünye denklemlerinin geliştirilmesini amaçlamaktadır. Fiber takviyeli kompozit ortamın anizotropik
bir yapıda olduğu varsayılmış ve piezoelektrik özelliğinden dolayı sıkışabilir olduğu kabul edilmiştir. Bunun
yanında fiber ailesinin uzamaz olduğu farzedilmiştir. Ayrıca kompozit malzeme fiber boyunca yön değişimine
duyarsız kalacağından, matematiksel olarak fiber vektörünün B→−B değişiminden etkilenmeyeceği için fiber
vektörünün bileşenlerinin dış çarpımı olan simetrik bir tansör tanımlanmıştır. Bu çalışmanın temelini elektrotermomekanik
denge denklemleri, fiber deformasyon geometrisi ve kinematiği ile ilgili denklemler
oluşturmaktadır. Bünye aksiyomlarının kullanılması sonucunda gerilme potansiyelinin Green deformasyon
tansörüne, fiber-dağılım tansörüne, elektrik alanı vektörüne ve mutlak sıcaklığa bağlı olduğu, ısı vektörü
fonksiyonunun ise bu büyüklükler ile birlikte sıcaklık alanının gradyanına bağlı olduğu görülmüştür. Kompozit
ortamın anizotropik yapıda olmasından dolayı, gerilme potansiyeli ve ısı vektörü fonksiyonları yaklaşık
teorilerden bulunmuş, etkileşimlerin tümü lineer kabul edilerek seri açılımı yapılmıştır. Uzaysal koordinatlarda
elde edilen simetrik gerilmenin, polarizasyonun ve ısı akısı vektörünün lineer bünye denklemlerinin ve denge
denklemlerinde yer alan ifadelerin denge denklemlerinde yerlerine yazılması sonucunda alan denklemleri
bulunmuştur.
1. Demirtürk S., 2011, “Fiber Takviyeli Piezoelektrik Kompozit Malzemelerin ElektroTermoelastik
Analizi”, Master Tezi, Süleyman Demirel Üniversitesi, Fen Bilimleri
Enstitüsü, Isparta.
2. Öntürk, N., 1993, “İki Fiber Ailesi ile Takviyeli Viskoelastik Kompozit Ortamlarda
Bünye Denklemlerinin Modellenmesi”, Doktora Tezi, Gazi Üniversitesi, Fen Bilimleri
Enstitüsü, Ankara.
3. Erdem, A. Ü., Usal. M. R., Usal. M., 2005, “Keyfi Fiber Takviyeli Viskoelastik
Piezoelektrik Bir Cismin Elektro-Termomekanik Davranışı İçin Matematiksel Bir
Model’’, Gazi Üniv. Müh. Mim. Fak. Dergisi, 20(3) 305-319.
4. Mikata, Y. 2000, “Determination of piezoelectric Eshelby tensor in transversely isotropic
piezoelectric solids’’, Int. J. Eng. Sci., 38, 605-641.
5. Dinzart, F., Sabar, H. 2009, “Electroelastic behavior of piezoelectric composites with
coated reinforcements: Micromechanical approach and applications’’, Int. J. Sol. and St.,
46, 3556–3564.
6. Tong, Z.H., Lo, S.H., Jiang, C.P., Cheung, Y.K. 2008, “An exact solution for the threephase
thermo-electro-magneto-elastic cylinder model and its application to piezoelectricmagnetic
fiber composites’’, Int. J. Sol. and Str., 45, 5205–5219.
7. Timoshenko, S.P., Goodier, J.N., 1970, “Theory of Elasticitiy’’, Mcgraw Hill, 567p.
8. Lubarda, V.A., 2004, “On Thermodynamic Potentials in Linear Thermoelasticity’’, Int. J.
Sol. Str., 41(26), 7377-7398.
9. Usal, M., Hamamcı, B. 2011, “A Mathematical Model for the Linear Constitutive
Equations of a Thermoelastic Composite Continuum Reinforced by Single Family of
Arbitrarily Fiber’’, J. Fac. Eng. Arch. Gazi Univ., 26(2), 315-324.
10. Maugin, A.G., Berezovski, A., 1999, “Material Formulation of Finite-Strain
Thermoelasticity and Applications’’, J. Ther. Stres. 22(4 and 5), 421-449.
11. Kalpakides, K.V., Dascalu, C., 2002, “On the Configurational Force Balance in
Thermamechanics, Proceedings’’, Math., Phy. and Eng. Sci., 458(2028), 3023-3039.
12. Eringen, A.C., 1966, “A Unified Theory of Thermomechanical Materials’’, Int. J. Engng.
Sci., 4, 179-202.
13. Mindlin, R.D., 1972, “Elasticity, Piezoelectricity and Crystal Lattice Dynamics’’, J.
Elasticity, 2, 217-282.
14. Tiersten, H.F., 1971, “On The Nonlinear Equations of Thermoelectro-Elasticity’’, Int. J.
Eng. Sci., 9, 587-604.
15. Nowacki, W., 1975, “Dynamic Problems of Thermoelasticity’’, Noordhoff International
Publishing, Netherlands.
16. Usal, M., Usal, M. R. Kurbanoğlu, C., 2009, “A Mathematical Model for the
Electrothermomechanical Behavior of Viscoelastic Piezoelectric Body Having
Hegzagonal Simetry’’, Sci. and Eng. Comp. Mater., 16, 153-171.
17. Usal M., Usal M.R, Erdem, A.Ü., 2009, “On Magneto-Viscoelastic Behavior of FiberReinforced
Composite Materials Part - I: Anisotropic Matrix Material’’, Sci. and Eng.
Comp. Mater., 16, 41-56.
19. Usal, M., Usal, M.R., and Esendemir, Ü. 2008, “A Continuum Formulation for Fiber -
Reinforced Viscoelastic Composite Materials with Microstructure Part-I: Anisotropic
Matrix Material’’, Sci. and Eng. Comp. Mater., 15(3), 217-234.
20. Usal, M.R, Usal, M., and Esendemir, Ü. 2006, “A Mathematical Model for
Thermomechanical Behavior of Arbitrary Fiber Reinforced Viscoelastic Composites–I’’,
Sci. and Eng. Comp. Mater., 13(4), 291-300.
21. Usal, M. 2010, “A Constitutive Formulation for the Linear Thermoelastic Behavior of
Arbitrary Fiber-Reinforced Composites’’, Mathem. Prob. Eng., (2010), Article ID 404398,
19 pages.
22. Usal, M. 2010, “Formulation of the Linear Constitutive Equations of Thermoelastic
Piezoelectric Materials’’, Electronic Journal of Machine Technologies, 7 (1), 13-30.
23. İnan, M. 1970, “Cisimlerin Mukavemeti’’, II. Baskı, Ofset Matbaacılık Ltd. Şti., İstanbul.
24. Spencer, A.J.M. 1984, “Continuum Theory of the Mechanics of Fibre Reinforced
Composites’’, Spencer, Springer Verlag, 284 p, New York.
25. Usal, M.R. 1994, “A mathematical model for the electro-thermomechanical behaviour of
fiber reinforced elastic dielectric media’’, Ph. D. Thesis, Erciyes University, Institute of
Science and Technology, Kayseri, pp: 108.
26. Hamamcı, B. 2006, “A mathematical model for fiber reinforced thermoelastic materials’’,
Master Thesis, Süleyman Demirel Univ., Institute of Science and Technology, Isparta,
pp: 93.
27. Eringen, A.C., Maugin,G.A., 1990, “Electrodynamics of continua: Foundations and Solid
Media’’, Springer-Verlag, New York.
28. Maugin, G.A., 1991, “Continuum Mechanics of Electromagnetic Solids’’, North –
Holland Series Applied Mathematics and Mechanics, Elsevier Scie. Netherlands.
29. Eringen, A.C., 1980, “Mechanics of Continua’’, Robert E. Krieger Pub. Co., Hungtington,
New York.
30. Şuhubi E.S., 1994, “Continuum mechanics–Introduction’’, İ.T.U., Faculty of Arts and
Sciences Publication. İstanbul.
M. Usal ve S. Demirtürk, “FİBER TAKVİYELİ PİEZOELEKTRİK KOMPOZİTLERİN ELEKRO-TERMOELASTİK ANALİZİ İÇİN MATEMATİKSEL BİR MODEL”, UTBD, c. 9, sy. 1, ss. 47–70, 2017.
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